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Question
The line segment joining the points A(3,−4) and B(1,2) is trisected at the points P(p,−2) and Q `(5/3,q)`. Find the values of p and q.
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Solution
Let P and Q be the points of trisection of AB.
Then, P divides AB in the radio 1:2
So, the coordinates of P are
` x= ((mx_2 +nx_1))/((m+n)) , y = ((my_2+ny_1))/((m+n))`
` ⇒ x = ({ 1 xx 1+2xx(3)})/(1+2) , y = ({1 xx 2+2xx(-4)})/(1+2)`
` ⇒ x = (1+6)/3 , y (2-8)/3`
` ⇒ x = 7/3 , y -6/3`
` ⇒x =7/3 , y =-2`
Hence, the coordinates of P are `(7/3, -2)`
But, (p -2) are the coordinates of P.
so, p = `7/3`
Also, Q divides the line AB in the ratio 2:1
So, the coordinates of Q are
`x = ((mx_2 +mx_1))/((m+n)) , y = ((my_2+my_1))/((m+n))`
`⇒x = ((2xx1+1xx3))/((2+1)) , y = ({ 2xx2+1xx(-4)})/(2+1)`
`⇒ x = (2+3)/3 , y = (4-4)/3`
`⇒ x = 5/3 , y =0`
Hence, coordinates of Q are `(5/3, 0)`
But the given coordinates of Q are `(5/3,q)`
so, q = 0
Thus, `p=7/3 and q =0`
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